Let me analyze the image for its mathematical content and provide solutions. PIL import Image import pytesseract

Load the uploaded image

image_path = "/mnt/data/17340515636321sz2xbz9.jpg" image = Image.open(image_path)

Use OCR to extract text from the image

extracted_text = pytesseract.image_to_string(image) extracted_textHere’s the extracted content from the image:

Q3 (K=4, C=1)
a) How many arrangements are there of the letters in the word COMPUTER?
b) How many of them begin with a consonant?

Q4 (A=4, C=2)
Determine the number of paths that spell PASCAL. Can combinations be used to solve each question?

Let me solve the questions systematically.

Q3a) How many arrangements are there of the letters in the word COMPUTER?

The word "COMPUTER" has 8 distinct letters. Therefore, the total number of arrangements is:
$8! = 40320$

Q3b) How many of them begin with a consonant?

The consonants in the word are C, M, P, T, R (5 consonants).

• If the first letter is a consonant, the remaining 7 letters can be arranged freely.
  Thus, for each consonant as the first letter, there are: $7! = 5040 \text{ arrangements.}$
• Since there are 5 consonants, the total number of such arrangements is: $5 \times 5040 = 25200$

Q4) Paths spelling PASCAL

The word "PASCAL" consists of 1 P, 2 As, 1 S, 1 C, and 1 L.

The number of unique permutations can be calculated by dividing the total arrangements of all letters by the factorial of identical letters:
$\frac{6!}{2!} = \frac{720}{2} = 360$

a) Combinations can be used to determine the number of arrangements, as you are selecting positions for identical letters (e.g., choosing 2 positions for A out of 6).
b) Using combinations simplifies identifying specific arrangements with constraints.

Would you like a more detailed explanation for any part?
Here are five related questions:

1. How would the answer change for Q3a if the word had repeated letters?
2. How many arrangements of "COMPUTER" end with a vowel?
3. How many arrangements of "COMPUTER" have all consonants together?
4. Can combinations be used to count paths for other arrangements of "PASCAL"?
5. What is the probability of randomly arranging "COMPUTER" such that it starts with a vowel?

Tip: When solving counting problems, identify unique vs. repeated elements early to simplify calculations.